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10/24/2013
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 1
PHY 113 C General Physics I11 AM – 12:15 PM TR Olin 101
Plan for Lecture 17:Review of Chapters 9-13, 15-16
1. Comment on exam and advice for preparation
2. Review3. Example problems
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 3
Webassign questions – Assignment #15
Consider the sinusoidal wave of the figure below with the wave function y = 0.150 cos(15.7x − 50.3t)where x and y are in meters and t is in seconds. At a certain instant, let point A be at the origin and point B be the closest point to A along the x axis where the wave is 43.0° out of phase with A. What is the coordinate of B?
xoo 7.15
18043
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 4
Webassign questions – Assignment #15
A transverse wave on a string is described by the following wave function. y = 0.115 sin ((π/9)x + 5πt)where x and y are in meters and t is in seconds.
(a) Determine the transverse speed at t = 0.150 s for an element of the string located at x = 1.50 m.
(b) Determine the transverse acceleration at t = 0.150 s for an element of the string located at x = 1.50 m.
tx
ttxy 5
9cos
9115.0),(
tx
ttxy 5
9sin
9115.0),( 2
2
2
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 5
Webassign questions – Assignment #15
A sinusoidal wave in a rope is described by the wave function y = 0.20 sin (0.69πx + 20πt)
where x and y are in meters and t is in seconds. The rope has a linear mass density of 0.230 kg/m. The tension in the rope is provided by an arrangement like the one illustrated in the figure below. What is the mass of the suspended object?
mgTk
c 69.0
20
tkxy sin0
T
mg
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 6
Comment about exam on Tuesday 10/29/2013
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 7
iclicker questionWhat is the purpose of exams?
A. Pure pain and suffering for all involved.B. To measure what has been learned.C. To help students learn the material.D. Other.
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 8
Advice on how to prepare for the exam
Review lecture notes and text chapters 9-13,15-16 Prepare equation sheetWork practice problems
Topics covered
Linear momentumRotational motion and angular momentumGravitational force and circular orbitsStatic equilibriumSimple harmonic motionWave motion
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 9
What to bring to exam:
Clear headCalculatorEquation sheetPencil or pen
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 10
iclicker question:Have you looked at last year’s exams?
A. Yes B. No
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 11
Linear momentum What is it? When is it “conserved”? Conservation of momentum in analysis of collisions Notion of center of mass
dtd
dtmd
dtdmm
m
pvvaF
pv
sNm/skg:momentumlinear ofUnits:momentum"linear "Define
if
f
i
f
i
ddt
ddt
pppFI
pF
:Impulse
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 12
Linear momentum -- continued
Physics of composite systems
ii
i
ii
i
ii
iii
ii dt
ddtmd
dtdmm pvvaF
:lawsecondsNewton'
ii
ii
ii
ii
ii
dtd
finalinitial
(constant)
0
: then,0if that Note
pp
p
p
F
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 13
Example – completely inelastic collision; balls moving in one dimension on a frictionless surface
i
iiv
iviv
vvv
vvv
pp
ˆ0.125
/5.03.0
ˆ15.0ˆ23.0
ˆ/1,ˆ/2
5.0,3.0For
21
21
21
2211
212211
finalinitial
m/s
sm
smsm
kgmkgmmmmm
mmmm
f
ii
iif
fii
ii
ii
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 14
Examples of two-dimensional collision; balls moving on a frictionless surface
--equationsmore2Need
,,,:Unknowns,,:Knowns
sinsin0
coscos
21
121
2211
221111
finalinitial
ff
i
ff
ffi
ii
ii
vvvmmvmvm
vmvmvm
pp
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 15
The notion of the center of mass and the physics of composite systems
2
2
2
2
2
2
:Define
:lawsecondsNewton'
dtdM
mMM
mdtmd
dtdmm
CMtotal
ii
ii
iii
CM
i
ii
i
ii
iii
ii
rFF
rr
rraF
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 16
Finding the center of mass
ii
iii
CM mMM
m rr
ji
jiir
jiir
ˆ00.1ˆ7504
ˆ22ˆ21ˆ)1)(1(
ˆˆˆ2;1:exampleIn this
321
332211
321
mm.
mmm
mmmymxmxm
kgmkgmm
CM
CM
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 17
Rotational motion and angular momentum Angular variables Newton’s law for angular motion Rotational energy Moment of inertia Angular momentum
dtddtd
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 18
Review of rotational energy associated with a rigid body
iii
iii
iii
iiirot
rmI
Irm
rmvmK
2
222
22
here w21
21
21
21
:energyRotational
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 19
Moment of inertia: i
iirmI 2
22MaI 22 22 mbMaI
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 20
22
21
21
:objectrollingofenergy kineticTotal
CM
CMrollingtotal
MvI
KKK
CMvRdtdR
dtds
dtd
: thatNote
22
222
21
21
21
CM
CM
CMrollingtotal
vMRI
MvRRI
KKK
22
21
21
:objectrollingofenergy kineticTotal
CM
CMrollingtotal
MvI
KKK
CMvRdtdR
dtds
dtd
: thatNote
22
222
21
21
21
CM
CM
CMrollingtotal
vMRI
MvRRI
KKK
CMCM
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 21
iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?
AB C
2MRI A 22 5.0
21 MRMRIB
22 4.052 MRMRIC
2
22
/12
01210
MRIghv
vMR
IMMgh
UKUK
CM
CM
ffii
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 22
How can you make objects rotate?
Define torque:
t = r x F
t = rF sin
r
F
αarτFraF
Imm
sinr
ατ I:motionrotationalfor lawsNewton'
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 23
Example form Webassign #11
X
t1
t3
t2
iclicker exerciseWhen the pivot point is O, which torque is zero?
A. t1?B. t2?C. t3?
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 24
Vector cross product; right hand rule
sinBACBAC
ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
i i j j k ki j j i kj k k j ik i i k j
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 25
From Newton’s second law – continued –conservation of angular momentum:
(constant)
00If
:Define
L
Lτ
prL
prτFr
dtd
dtd
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 26
Example of conservation of angular momentum
wheelbf
wheelwheelbf
wheelibiwheelfbf
LLLLL
LLLL
20
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 27
Summary – conservation laws we have studied so far
Conserved quantity Necessary conditionLinear momentum p Fnet = 0Angular momentum L tnet = 0
Mechanical energy E No dissipative forces
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 28
Fundamental gravitational force law and planetary motion Newton’s gravitational force law Gravity at Earth’s surface Circular orbits of gravitational bodies Energy associated with gravitation and orbital motion
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 29
Universal law of gravitation Newton (with help from Galileo, Kepler, etc.) 1687
212
122112
ˆrmGm rF
2
211
kgmN10674.6
G
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 30
Gravitational force of the Earth
RE m
2226
2411
2
2
m/s8.9m/s)1037.6(
1098.51067.6
E
E
E
E
RGMg
RmGMF
Note: Earth’s gravity acts as a point mass located at the Earth’s center.
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 31
days27.4s32367353.951098.51067.6
)1084.3(π2
π2
π2ω
:Earth todueMoon for lawsNewton'
2411
38
3
2
2
E
EM
EMEM
EM
E
EM
MMM
GMRT
RT
Rv
RGM
Rva
M aF
Stable circular orbit of two gravitationally attracted objects (such as the moon and the Earth)
REMF
a
v
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 32
m1
R2
R1
m2
v1
v2
Circular orbital motion about center of mass
CM
21
321
21
2
111
1
2
1
11
1
21
1
2211
2
22
2221
21
1
21
1
2
212
mmGRRTT
TRm
RTRm
Rvm
RmRmRvm
RRmGm
Rvm
2
31
21
321
21
1212
22
then if that Note
GmR
mmGRRTT
RRmm
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 33
(const)
0
L
Lτdtd
m1
R2
R1
m2
v1
v2
L1=m1v1R1
L2=m2v2R2
L = L1 + L2
2
2
1
1
RL
RL
Note: More generally, stable orbits can be elliptical.
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 34
Gravitational potential energy
rmGm dr
rmGmrU
rmGmdrU
r
gravity
r
rgravity
ref
212
21
221
''
)(
ˆ)(
rFrF
hRmGMhRrU
E
SEEgravity
)(
Example:
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 35
Analysis of static equilibrium
Meanwhile – back on the surface of the Earth:
Conditions for stable equilibrium
0: torqueofBalance
0:forceofBalance
ii
ii
τ
F
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 36
0)()2(:Torques 1 CMg RmgmF
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 37
T
Mgmg
**X
2/
x
t
sin2//
0sin2
0
MgmgxT
TMgmgx
NTNMgNmgmxmo
313200600
2853For
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 38
Some practice problems
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 39
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10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 40
From webassign:
A 100-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force would have to be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.800 rev/s in 2.00 s? (State the magnitude of the force.)
FR
view from top:
2
21 MRI
I
αFrτ
10/24/2013 PHY 113 C Fall 2013 -- Lecture 17 41
From webassign:A 10.3-kg monkey climbs a uniform ladder with weight w = 1.24 102 N and length L = 3.35 m as shown in the figure below. The ladder rests against the wall and makes an angle of θ = 60.0° with the ground. The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is connected to the wall by a horizontal rope that is frayed and can support a maximum tension of only 80.0 N.